This essay is full of mistakes. Idea after idea and sentence after sentence is simply wrong. This sentence, for example, is false. Worse yet, this not even complete sentence! A long time ago (so the legend goes) a Cretan prophet by the name of Epimenides declared that “All Cretans are liars.” This paradoxical statement has come to be known as the Epimenides paradox or the Liar paradox This Adam (or atom) of paradoxes has been reformulated into countless variants, yielding such gems as “I am lying,” and “this sentence is false.” It has been split, (“The following sentence is true. The preceding sentence is false.”) boxed, translated and quoted in the Bible. In short, one would assume that the Liar Paradox had been beaten to death. In 1931, a German mathematician named Kurt Gödel breathed new life into the Liar paradox in a paper poetically entitled “On Formally Undecidable Propositions in Principia Mathematica and Related Systems I”: Gödel’s work demonstrated that paradox forms an implicit part of every axiomatic system of logical reasoning. In this essay, I will be examining the problems which self reference and paradox pose to systems of reasoning especially formalized mathematical and logical reasoning. These two areas, in their quest for objective truth become very interesting in the light of Gödel’s revelations. In the end, it may turn out that their quests for a formalized objective truth may have been in vain. In addition, I will sometimes be referring to myself (with good reason).

Before we proceed, it is vital that we understand why self-reference is such an important topic. We must ask, “why write this essay at all?” Of course, in asking that question, I have answered it. Self-reference is one of the most powerful aspects of human thought and language. On the one hand, it is very useful, allowing me to refer to the subject of this essay (self-reference and Gödel) and even myself. How could I survive if I could not complain that “My arm has been cut off and I’m going to die”? On the other hand, as we have already seen, self-reference also allows us to construct such sentences as “I am lying”, or “ignore these words”. Perhaps most troubling would be constructs like the following one, which proposes

Let us make a new convention: that anything when enclosed in triple quotes- for example ”’No, I have decided to change my mind; when the triple quotes close, just skip directly to the period and ignore everything up to it”’ -is not even to be read (much less paid attention to or obeyed).

To a logician or a mathematician who is attempting to construct a system which is clear and precise, the possibility of such a statement is most distressing. In order for a system to be truly reflective of reality, conventional math wisdom goes, it must be able to accurately represent all truths while side stepping paradox. Over the years, various mathematicians have attempted to create a system free from paradox and they have gone about it in a variety of ways. It will perhaps be helpful to look at this development.

A good paradox for beginning our discussion of mathematics is Grelling’s paradox. If one wished, one could divide all of the English adjectives into two categories: autological and heterological. An autological adjective is one which is self-descriptive; “pentasyllabic”, “awkwardfulness”, “short”. A heterological adjective is not self-descriptive; “edible”, “hungry”, “monosyllabic”, “long”. Now, the question we must ask ourselves is: “Is ‘heterological’ heterological?”

This idea is analogous to some problems with the theory of sets first developed by Georg Cantor in the 1880’s. In the 1900’s, the mathematician Gottlob Frege attempted to secure the foundations of number theory in logic, using set theory. His project was to clearly define the natural numbers, ensuring that our understanding of them was free from contradiction and paradox. His attempt was doomed to failure because of a paradox implicit in set theory. A set can be either a member of itself, or not. Most sets are not members of themselves: the set of cabbages is not a cabbage, the set of Luke Skywalker is not Luke Skywalker. On the other hand, some sets do contain themselves: the set of all sets, the set of things which are not paintings. Now, we can easily imagine a set S: the set of all sets which are not self-containing. Recalling Grelling’s paradox, we now ask “is S a member of this set?” This is called Russell’s paradox as it was discovered by Bertrand Russell, spelling the downfall of Frege’s system.

Once Russell had discovered this problem, he examined mathematical reasoning and concluded that the paradoxes came from self-reference. In order to eliminate paradox, Russell thought, it is necessary to eliminate all forms of self-reference. This task proves rather difficult due to the fact that self-reference is not always contained in one step statements. In addition to banning “This statement is false” Russell also had to deal with two or more step paradoxes such as the one already witnessed (“The following sentence is true. The preceding sentence is false.”). In order to realise this goal, Russell had to figure out how to ban both direct and indirect self-reference. Ultimately, Russell, like Frege wanted to create a system powerful enough to be able to prove all truths about the natural numbers, without falling into the inconsistency of paradox. He was looking for a system which was complete and consistent.

The result of this search was a four volume work, the Principia Mathmatica, published between 1910 and 1913 by Russell and Alfred North Whitehead. Their goal was to create a system of logical and mathematical reasoning which was impervious to paradox. They introduced into mathematics a strict system of hierarchical types where a set of one type was only allowed to refer to sets of lower types. The lowest order could only contain “objects” and no sets. The next type could contain only objects and the lowest order of sets. Thus no set could contain itself. Set S would be forbidden. In theory, this also dealt with the two step Liar paradox in that neither (or each) sentence is of a higher type than the other, so it must be meaningless (ie; together they cannot be formulated within the hierarchical system). However, there are still many intuitive problems with this system.

As I mentioned before, self-reference is an important part of daily life. Under Russell’s system, I could not refer to this essay as I am now doing, I would need a higher order essay, a meta-essay if you will. Worse, I could never refer to myself at all! In fact, the theory of types, if applied to language is a violation of itself. How can one argue that the Liar paradox is meaningless? To what level of type does the word ‘meaningless’ belong? The Principia Mathematica, in its zeal to eliminate paradox, does away with several useful concepts which are important parts of human reasoning. With no recourse to references of self, we are left with no way of expressing anything about ourselves. Ironically, even these drastic sacrifices for the sake of avoiding paradox fail.

Just when Russell and Whitehead thought that they had created a fortress against the evils of self-reference, Kurt Gödel found a chink in the wall. In his 1931 paper “:On Formally Undecidable Propositions in Principia Mathematica and Related Systems I” he makes the following proposition (number VI)

“Actually, it was in German, and perhaps you feel that it might as well be in German anyway. So here is a paraphrase in more normal English:

All consistent axiomatic formulations of number theory include undecidable propositions. This is the pearl.”

Gödel’s theorem is proven by reversing Russell and Whitehead’s work. Where Russell and Whitehead showed that numbers could be mapped isomorphically onto logical statements, Gödel showed that logical statements could be mapped isomorphically onto numbers. A way of understanding how this mapping is possible might be to consider the analogy of feedback loops created with electric guitars.

When the string of an electric guitar is plucked, the vibrations induce an electrical current which causes a magnet to induce vibrations in a speaker. The speaker in turn causes vibrations in the air which can cause the string to resonate, causing more electrical signal, causing more sound, causing more resonance and so on. The vibrations are mapped on to an electrical current which is mapped on to vibrations.

Instead of creating a feedback loop, Gödel’s theorem creates a paradox (like a guitar which, when played through a particular amplifier, produces a sound whose vibrations actually destroy the guitar and the amplifier). More specifically, Gödel used his mapping system to create a mathematical statement G which means “I cannot be proven in system X.” (where X is the system in which G is written). This statement destroys any hope of completeness in system X. Because G is a true statement, it is unprovable, therefore, system X is incomplete; it is not powerful enough to capture all truths.

“It sounds a bit like a science fiction robot called ‘ROBOT R-15′ droning (of course in a telegraphic monotone):

ROBOT R-15 UNFORTUNATELY UNABLE TO COMPLETE TASK T-12 -VERY SORRY

Now, what happens if TASK T-12 happens, by some crazy coincidence, to not be the assembly of some strange cosmic device but merely the act of uttering the preceding telegraphic monotone?”

In other words, in order for the robot to perform the task, it has to be blatantly self-contradictory. A self-contradictory robot (or formal system) is pretty much useless. Therefore it must be unable to perform the task (prove the unprovable) and thus incomplete. Paradox! Gödel goes on to prove that this must always be true of any system of reasoning, no matter how powerful.

The only question left to us is whether math can recover from this terrible blow. Unfortunately for school children everywhere, it can. The eternal presence of paradox is no more of a problem for math than it is for English. One can still find and use all sorts useful expressions, or truths. However, the fact that we cannot create a formal system which can capture all of mathematical truth casts serious doubt on the objectiveness of such truth. It may well be that in math there are multiple truths, some of which are contradictory (an instance of this can be found in the difference between euclidean and non-euclidean geometry (or any other undecidable proposition)). “Thus, somewhat counter intuitively, it turns out that mathematical reasoning has no fixed and eternal boundaries either.”

Ever since Epimenides, declared that “All Cretans are liars,” the problem of paradox has been at times deep and troubling one, at times annoying and at times very enlightening. By its nature, it leads us to the fundamental problem of knowledge, namely, “is it possible to formally codify the Universe in such a way that our system of coding will be both complete and consistent?” Strangely enough, the answer is no.